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Christoffel Symbols

Consider a region in which the metric field is g_ab. Christoffel symbols have a vital role in calculating the effect of parallel displacement.


Christoffel symbols, defined using partial differentiation, are not tensors, but indices can be raised and lowered in the usual way. It is common to raise the first index.


Clearly Christoffel symbols are symmetric in the last two suffixes, and satisfy
Christoffel symbols are sometimes said to define the connection in general relativity. From an empirical perspective, the connection is the physical prescription for comparing the coordinate system set up an observer with that of a second, nearby observer, using parallel displacement in tangent space. In pure mathematics there is no physical prescription, so a different definition of a connection is required. In my view, adopting a mathematical definition reduces the empirical foundations of general relativity to metaphysics, in conflict with Einstein’s approach to physics. In this empirically based account, the purpose of Christoffel symbols is to enable us to calculate the effect of parallel displacement without reference to a non-physical tangent space. A tangent chart at x is defined with non-physical metric h using primed coordinates. At x,
Observe that, from Clairaut’s Theorem, the partial derivative of the transformation matrix is symmetrical in its lower indices,
Interchange a ↔ c and b ↔ c,
Then,

Parallel Displacement

Parallel displacement of a vector p from x to x + dx in tangent space keeps the primed components constant, . Multiply by , to lower the index and convert to unprimed coordinates,
Then, ignoring terms O(max(dxⁱ)),
Substituting the Christoffel symbol eliminates the dependency on tangent space,
Raise and lower indices to find the standard formula for infinitesimal parallel displacement referring to covariant components,


For a second vector q, p · q is invariant,
Since this is true for all p_a, we find the standard formula for infinitesimal parallel displacement referring to contravariant components,

The Covariant Derivative

We have seen that the partial derivative of a vector field is not a tensor field because its definition requires a subtraction between vectors in different vector spaces. However, we may parallel displace a vector from x to a nearby point, x + dx and it becomes a vector defined at x + dx. Then we can define the covariant derivative using vector addition, and the result will be a tensor.


One may see that i is a vector index from first principles, or by using the result that the partial derivative of a scalar field is a vector field for each value of a and b. Evidently,


The covariant derivative of the tensor product p^aq^b may be found from first principles by parallel displacing p^a and q^b. One finds
The product rule of differentiation holds,
Since these rules apply to a basis, they apply to all contravariant rank 2 tensor fields by linearity. Similarly,
One may extend this rule to tensors with any number of up and downstairs indices, adding a Γ term for each superfix and subtracting one for each suffix (use induction). The rule applies also to scalar fields, and states also that the covariant derivative of a scalar field, S : x → S(x), is equal to the partial derivative,
The second covariant derivative of a scalar field, S, commutes,
The metric behaves as a constant with respect to covariant differentiation;
This simply states that vectors have constant magnitude under parallel displacement.

Geodesics

A geodesic is a curve found by parallel transport of a vector in the direction of that vector. To apply parallel displacement to the tangent vector, , of a curve, t → x^a(t) we put into the standard formula for infinitesimal parallel displacement referring to contravariant components,
Thus, we obtain:


For a time-like geodesic, we may normalise the tangent vector. In this case, the parameter t is proper time. We may rewrite the geodesic equation in terms of velocity, ,
If a particle is represented by a density, rather than as a point, then velocity is a field, and we have,
Then the geodesic equation takes the simple form:


In a reference frame in which the particle is not moving, v^b = (1, 0, 0, 0), and this simply states
i.e. proper acceleration is equal to zero. This holds in any frame, since proper acceleration is a vector (the derivative of a vector with respect to a scalar). Using τ for proper time,
We can now make the distinction that an active force causes a proper acceleration according to Newton’s second law, while an inertial force does not. An active force can thus be considered as a physical quantity represented by a vector, while an inertial force is frame dependent.

The Riemann Curvature Tensor

In order to describe the curvature in spacetime we require a tensor field defined in terms of second derivatives, with four indices, and which vanishes in flat space. The second covariant derivative of a vector field, p_a : x → p_a(x), is
If we antisymmetrise the second and third indices, by switching them and subtracting, a number of terms drop out.
This is a tensor equation for all vectors p_c. So, by the quotient theorem,
is a tensor field. is known as the Riemann curvature tensor, and depends only on the metric and the connection. We have that the antisymmetrised second covariant derivative of a vector field is equal to curvature tensor contracted with that vector field,

In a flat space we may choose rectilinear coordinates in which the Riemann curvature tensor vanishes. Contrariwise, if the Riemann curvature tensor vanishes, we may parallel displace a vector p_a from x to x + dx, and then to x + dx + dy and the result is the same as if we parallel displaced it from x to x + dy and then to x + dy + dx. So, parallel transport is path independent. Rectilinear coordinates may then be defined by parallel transport of an orthonormal axes set to any point, showing that space is flat.

Symmetries of the Curvature Tensor

It is easily seen that
and
On lowering the top index,
Using
we find
Using
we find
Then examination reveals the symmetries,
and
As a result of all these symmetries, one may calculate that the Riemann curvature tensor has only 20 independent components.

The Bianchi Identity

To find the second covariant derivative of a tensor field, first consider the outer product of two vector fields p_a and q_b,
Interchange i and j, and subtract,
The antisymmetrised second covariant derivative of a vector field is equal to curvature tensor contracted with that vector field. So,
So, by linearity, for a rank 2 tensor field T_ab,
Now, if T is the covariant derivative of a vector field, T_ab = p_a;b,
Make cyclic permutations of b, i, j and add, noting that ,
This is true for all vectors p_c. so we have the Bianchi identity,

Contracted Curvature Tensors

If we contract the top index with the mid lower we find a symmetrical tensor,
known as the Ricci tensor. (Older treatments usually contract with the last index. This is an opposite sign convention due to antisymmetry of the last two indices, and results in a minus sign which appeared in Einstein’s original formulation of the field equation, but which is not usual in more recent accounts). Contracting again gives the scalar curvature, total curvature, or Ricci scalar,
The Bianchi identity has five indices. If we contract it twice, we will have a relation with one suffix. Put c = j and multiply by g^ab,
Because g behaves like a constant with respect to covariant differentiation, we can use it to contract under the derivative. Since the Ricci tensor is symmetrical, it can be written with one index above the other. Thus,
Now
So,
This is the Bianchi identity for the Ricci tensor. Raising the suffix i,
prompts the definition of the Einstein tensor,
Then the contracted Bianchi identity can be written more neatly:


Riemann Curvature ↑ Einstein’s Law of Gravitation →